Who decides who goes first? You do - based upon your actions. There are many times throughout a fencing bout when no one has Right of Way. For example - neither of you are taking the initative (you're both either being lazy, or patient) or both of you are taking the initative (you're bothing charging - this is usually ruled a simultaneous action. In the end though, after skirting around one another for a bit of time, someone is going to come out with the attack - initating an offensive move when your opponent does not claims Right of Way. It's a simple idea, but it takes a while to train your mind to react in the proper fashion.

My opponent has Right of Way. What do I do?
Take it away from him/her. This is key to fencing foil or sabre. There are three main ways to get Right of Way back after losing it:

Defense: Here you have two options - parry and dodge. Either of these options works equally well in theory and depending upon your style and ability you'll hopefully be flexible enough to use both - deciding at the necessary moment which to rely on.

Counter Offense:There are a range of more or less successful actions that fall into this category. A counterattack is not usually a wise idea. If you just lost Right of Way through a miss or parry you can always Remise. A stop cut is available in sabre. If your opponent is still some distance away and hasn't fully committed him/herself yet, Point in Line is an option. But: Battement is the typical solution that one chooses. Again - these are either attacks against your opponent or against his/her blade while they have Right of Way - if they hit you, you will lose the point even if you hit them too.

Attack - Parry - Riposte - Remise In this case, fencer A attacks, is parried by fencer B, fencer B then initiates an attack (a riposte) (because it's his turn) and instead of defending that attack, fencer A chooses to strike at fencer B again. In order to visualize this sequence, it is important to realize that fencers are trained not to pull back from your opponent when parried - it is easier to defend when your weapon is already extended. So, the action would look like this - fencer A strikes at fencer B's belly. (Attack) Fencer B turns his bell guard to his left and moves his hand all of four inches to allow fencer A's blade to land harmlessly on his guard. (Parry) Fencer B then forgets about Fencer A's blade lying a scant few inches away from belly, extends his weapon, and smacks Fencer A in the face. (Attack) Fencer A carries on with his attack, even though parried, and hits Fencer B in the stomach. (Remise) Again, the idea is that only Fencer B scored a hit that would be damaging or fatal. Not because of location of hit, but rather because the force of fencer A's action was stopped by the fencer B's parry, and the Remise itself would not have been powerful enough to cause harm.